Function spaces in complex analysis VT23

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• Function spaces in complex analysis VT23

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  Lectures:

  
  

  When:  Mondays 13:15-15:00  (start 30/1)
  

  Where: Albano House 1, 3rd floor
  30/1: Kovalevsky rummet
  

  from 6/2: Cramérrummet

  
  

  Teachers: Annemarie Luger and Salvador Rodriguez Lopez
  

  Examination: presentation and oral exam

  Prequisitaries:
  - Some complex analysis
  - Some measure theory
  - Interest in Analysis in general!
  

  • 

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• Schedule for presentations

  
  

  3/4      Christian and Linus (Bergmanspaces)
  
  17/4    usual lecture
  

  24/4    Nedialko and Ludvig (Hankel operators)
  

  8/5     Lefteris (duality theorem) 
             Dario (Toeplitz operators)
  

  15/5   Ellen (Maximal function)
            Vlad (Privalovs article)
  

  22/5  finishing usual lecture
  

• Lecture 6/3
  • We used the pointwise majorisation result that we had for harmonic extensions of $L^1(\mathbb{T})$ functions.
  • We gave some mild conditions for convolution operators to be poitwise majorised by the Hardy-Littlewood Maximal function. This automatically applies to the Poisson integrals.
    
  • We introduced the weak $L^p$ spaces $L^{p,\infty}$, and the distribution function.
    
  • We expressed the $L^p$ norm of a function in terms of its distribution function.
    
  • We stated, for $p\in (1,\infty)$, the $L^p$ boundedness of the Hardy-Littlewood function, as well a the $M:L^1\to L^{1,\infty}$ boundedness result.
    
  • Assuming the $M:L^1\to L^{1,\infty}$ boundedness, we proved that $M:L^p\to L^p$ for $p\in (1,+ \infty]$. We gave an interpolation proof.
    
  • We stated the Marcinkiewicz interpolation theorem for sub-additive operators $T$ (If you revise the proof we gave for $M$, we proved Marcinkiewicz theorem under the diagonal case $T:L^\infty \to L^\infty$ and $T:L^1\to L^{1\infty}$).
  • As a consequence of the general Marcinkiewiz result, we obtain Young's/Haussdorf inequality for $1$, and announced that for $p\in (0,1]$, the same inequality holds provided we change $L^p$ by $H^p(\mathbb{D})$. The result for $p\in (0,1]$ is due to Hardy and Littlewood (1927).
    

  
  

  References
  

  Zo's Lemma( More general than the one we established during the lecture) http://matwbn.icm.edu.pl/ksiazki/sm/sm55/sm5518.pdf

  
  

  Chapter VIII (A,B1, B2)
  

  Koosis, Paul Introduction to Hp spaces. Second edition. With two appendices by V. P. Havin [Viktor Petrovich Khavin]. Cambridge Tracts in Mathematics, 115. Cambridge University Press, Cambridge, 1998.

  
  

  Notes:
  

  1)The proof of the weak $L^1$ inequality uses the so-called "Rising sun Lemma". This will differ from the approach in the lectures.
  

  2) The Maximal function in the book is the so-called uncentered Maximal function, while the one in the lectures is called the centered maximal function. On can indeed show that both are pointwise equivalent to each other (Exercise).
  

  
  

  General interpolation theory:

  Bergh, Jöran; Löfström, Jörgen Interpolation spaces. An introduction. Grundlehren der Mathematischen Wissenschaften, No. 223. Springer-Verlag, Berlin-New York, 1976.
  

  Bennett, Colin; Sharpley, Robert Interpolation of operators. Pure and Applied Mathematics, 129. Academic Press, Inc., Boston, MA, 1988.

  
  

  
  

• Lecture 13/3

  • We motivated the study of Maximal operators (I called the Fundamental Analysis principle [as it appears in Krantz' book]), as a natural way to study pointwise convergence. In some cases, the pointwise convergence is indeed equivalent to the boundedness of associated maximal operators (Stein's principle).
  • We introduced  the notion of Subharmonic Functions, and established some properties.
  • We gave an equivalent characterisation, for continuous functions for being harmonic (We only proved one direction. The other can be found in the book of Duren).
  • We deduced the property that averages on spheres of radius r of subharmonic, continuous non-negative functions is non-decreasing  in r.
  • We stated Jensen's formula (w/o proof).
  • Obtained as a corollary that given a holomorphic function in the disc then $\log |F|, \log_+ |F|$ and $|F|^a$ for $a>0$ are subharmonic functions. 
  • We (finally!) give the definition of Hardy spaces $H^p(\mathbb{D})$.    

  References

  Duren, Peter L. Theory of Hp spaces. Pure and Applied Mathematics, Vol. 38 Academic Press, New York-London 1970 xii+258 pp
  

  Krantz, Steven G. A panorama of harmonic analysis. Carus Mathematical Monographs, 27. Mathematical Association of America, Washington, DC, 1999. xvi+357 pp.
  

• Lecture 20/3
  • We recalled the theorem regarding the existence of non-tangential limits of functions in $h^p$ for $p\in (1,\infty]$.
    
  • We showed that if $p\in (0,1]$., those properties are preserved for functions in $H^p$ provided they have no zeroes.
  • We proved Jensen's formula and the accompanying lemma stated in the previous lecture
  • We proved the completeness of Hardy spaces a metric spaces.
  • We proved that the zeroes of functions in the Nevalinna class can't be too far away from zero, so we can form a Blaschke product with them.
  • That Blaschke product $B$ is an element in $H^\infty(\mathbb{D})$, with $\Vert B\Vert\leq 1$. Hence  the non tangential limits exists.
  • (We stated without proof) that the Blaschke product $B$ satisfies that $|B(\xi)|=1$ a.e. and $\lim_{r\to 1} \int_{\mathbb{T}} \log |B(r\xi)| d \xi=0$.
  • We stated the first factorization theorem: If $X=N$ or $H^p$, and $F\in X$ is not identically zero, then if $B$ is the Blaschke product formed with the zeroes of $F$, then we can write $F=BH$ with $H\in X$ and $\Vert F\Vert_X=\Vert H\Vert_X$, and $H$ has no zeroes.
    
  • We gave the proof for $X=N$.
    
    

• Lecture 27/3